Timeline for Function Spaces on the Open Unit Disk defined by Hardy Space norms
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 21, 2020 at 20:21 | comment | added | Yemon Choi | It is worth noting that while I agree with @ChristianRemling's observation that what you have defined can be viewed as a vector-valued $L^\infty$-space, the result will not be a Hilbert space, and it will lose all the important connections between e.g. harmonicity/holomorphicity and the Hardy space. Note that if $g$ belongs to this larger space there is no reason to suppose its behaviour on the circle of radius $1/2$ in any way controls what happens inside that circle, and $g$ need not have any kind of power-series expansion | |
| Dec 21, 2020 at 15:48 | comment | added | Christian Remling | @MCS: No, sorry, I don't know of a reference that discusses this topic in detail. The proof of the completeness of $L^{\infty}((0,1); X)$, $X$ a Banach space, should be essentially the same as in the classical case $X=\mathbb C$ (if $f_n$ is a Cauchy sequence, then $f_n(x)$ is a Cauchy sequence in $X$ for all $x\in (0,1)\setminus N$, $|N|=0$, so the pointwise limit $f(x)=\lim f_n(x)$ exists, and then show that $f_n\to f$ in norm). | |
| Dec 21, 2020 at 1:36 | comment | added | MCS | Wonderful! Is there a name or any keywords associated to Banach spaces whose elements are Banach-space-valued functions of a real variable? That is, what should I search for to find information about this sort of thing? (For example: a proof that this is a Banach space?) | |
| Dec 20, 2020 at 23:49 | comment | added | Christian Remling | I think you can tie this up with mainstream theory by viewing $g$ as a function $g: (0,1)\to L^2(S)$, and now of course $g(r)$ is the restriction of your $g$ to the circle $re^{i\varphi}$. So I think the usual notation for your space would be $L^{\infty}((0,1); L^2(S))$ (and yes, this is a Banach space). | |
| Dec 20, 2020 at 1:40 | comment | added | Yemon Choi | Thanks. Upon re-reading, perhaps I was a little hasty with my second comment; there is a notion of "harmonic Hardy space" which is by definition a space of real-valued harmonic functions on the disk satisfying the growth condition that you refer to as the space-defining norm. I think what is going on here is that "(1-|z|) times holomorphic" is still much more special than just continuous | |
| Dec 20, 2020 at 1:25 | comment | added | MCS | Thank you for this comment. I have changed the title, as per your suggestion. Your comment is in part exactly why I asked this question. Now I know that what I'm working with doesn't fall under the provenance of classical Hardy Space theory. | |
| Dec 20, 2020 at 1:24 | history | edited | MCS | CC BY-SA 4.0 |
edited title
|
| Dec 20, 2020 at 0:50 | comment | added | Yemon Choi | In particular, "the space-defining norm" is not the characteristic feature of Hardy spaces; the holomorphic structure (and the ensuing boundary behaviour as $r\nearrow 1$) is the crucial factor. | |
| Dec 20, 2020 at 0:48 | comment | added | Yemon Choi | Would you mind editing your title? Hardy spaces on the disk have established definitions, and if you want to consider larger spaces then you really really really really should not talk about "atypical elements of Hardy spaces" | |
| Dec 19, 2020 at 23:17 | history | asked | MCS | CC BY-SA 4.0 |